1. Multi-item Location Inventory Models
Evren Sarınalbant

2. Contents Facility location models Location-inventory models
Multi-item concept
Integration of multi item to L-I
Conclusion

3. Facility Location Models
Models considering location of new facilities subject to transportation costs
Most basic form of location problems
Minimize the total distance between new facility and the existing ones

4. Facility Location Models
New Facility

5. Facility Location Models
Distance calculations
Rectilinear distance
Euclidian distance
Location-allocation models determine which existing locations will be served by the new facility

6. Facility Location Models
Location on networks aims to locate one or more facilities on a network
Networks consist of predetermined candidate points and/or the routes between them
(Nahmias 2001)

7. Location Inventory Models
An extension of location allocation models
Integrate:
Fixed cost of opening a new facility
Transportation costs

8. Location Inventory Models
Cost of holding safety stock
Generally, location of distribution centers in a three-tiered supply chain
Determining optimal number,location of DCs and the allocation of retailers to these DCs for single product

9. Location Inventory Models

10. Location Inventory Models
Why Location Inventory?
Importance of logistics has enhanced
Companies aim to reduce transportation and inventory costs to gain competitiveness
Centralization is proved to be cost effective (Eppen 1979)
Customer responsiveness affects the market share of the firms

11. Location Inventory Models
Location-allocation models come up with intuitively suboptimized solutions since inventory costs are not considered
Inventory control sytems are used after the design of supply chain. Lee and Billington (1992) underlined that a common pitfall in SCM is the seperation of supply chain design from operational decisions

12. Location Inventory Models
Relevant literature studies on L-I
Nozick and Turnquist (1998) considered integration of inventory impacts into a fixed-charge model for locating DCs. They calculated inventory levels and stockout rates for multiple DCs in order to catch a linear trend. Then, incorporated these impacts into the model.

13. Location Inventory Models
Erlebacher and Meller (1999) constructed a location inventory model. Their model is a detailed one consisting of fixed costs of opening DC, inventory cost of single item (safety stock) and transportation costs. They use squared Euclidian distance metric. They determine the optimal number and locations of DCs by using some heuristics

14. Location Inventory Models
Daskin et al (2002) constructed an inventory location model which determines the optimal locations of DCs from a set of candidate points and solved by using Lagrangian relaxation embedded in branch&bound

15. Multi Item Concept
In recent years, it is realized that road to survival passes through variability.
Therefore, inventory control in a multi-item environment became more important to achieve a good customer responsiveness and reduction in costs

16. Multi Item Concept
In the location-inventory models constructed upto now, single-item inventories were considered.
However, in design of supply chains, multi-item systems are frequently encountered. The issues that may affect the design of system in multi-item environments are:

17. Multi Item Concept
Intercorrelated product demands, meaning an increase in demand of one product can increase or decrease the demand of others (positive-negative correlation)
Ex: Demand of harddisk drives affects the demand of mainboards (positive correlation)
Ex: Demand of CD-writers affects the demand of large size harddisks (negative correlation)

18. Multi Item Concept
Hausman et al (1997) considered multi item inventory systems and analyzed joint-demand fulfillment probability with respect to correlated demands of items.
They used a multivariate normal distribution for the demands and done a probabilistic analysis for no stockouts to determine order-up-to level for each item

19. Multi Item Concept
Autocorrelation, meaning demand of an item in one period can affect the other period
Ex: In durable goods sector, high demand in one period may cause it to decrease in the next period
Priority of items, some items may be more important for distributor. (more profit margin, more sensitive customers)

20. Multi Item Concept
The holding and shortage costs may change from item to item. The penalties which are incurred by shortage of particular items may be larger than the other. (because of supply contracts)
Some items need to be kept under special conditions, so that they can affect the location of DC.

21. Integration of Multi-item with L-I
By looking at the previous information, it is beneficial to design the supply chain by using location inventory models considering multiple items.
In order to accomplish this, one must integrate demands of these items into the objective function

22. Integration of Multi-item with L-I
The problem is constructing a model which incorporates multi-item inventory costs and the factors that may affect the optimal location of DCs in a supply chain where supplier’s and retailers’ locations are known and fixed.

23. Integration of Multi-item with L-I
In the study of Daskin et al,

24. Integration of Multi-item with L-I

25. Integration of Multi-item with L-I
In order to reach to multiple item case, a multivariate normal distribution can be applied for each item demand by the help of the study done by Hausman et al.

26. Conclusion
Location-inventory models has taken attention in OR recently.
Therefore, there are few papers in the literature which analyzed this issue.
After my literature search, there seems to be no studies on multi-item location inventory
A detailed analysis should be done on this subject

27. Questions&Comments